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Resurgence number and convex body associated to pairs of graded families of ideals

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Pith's one-line read Asymptotic resurgence of graded families of ideals is a convex-body dilation threshold, proved for monomial and Rees-package ideals.

desk verdict A promising but under-supported main theorem: the proofs conflate integral closures with the ideals themselves, and Theorem 3.6 needs a normality hypothesis or a restatement in terms of integral-closure resurgence. read the letter →

arxiv 2412.04417 v1 pith:ZRDF6UPR submitted 2024-12-05 math.AC

Tai Huy Ha , A.V. Jayanthan , Arvind Kumar , Thai Thanh Nguyen This is my paper · ORCID
classification math.AC MSC 13A1813F2013A30
keywords asymptoticresurgencegradedfamiliesofidealsNewton-OkounkovbodymonomialReespackagedeterminantalintegralclosureconvexbodies
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper seeks to show that the asymptotic resurgence number of a pair of graded families of ideals—the supremum of $s/r$ for which $a_{st}$ fails to be contained in $b_{rt}$ for all large $t$—is encoded in convex geometry. For monomial ideals, the paper proves that, when the Rees algebra of the second family is Noetherian, $\hat{\rho}(a_\bullet,b_\bullet)$ equals the largest $\lambda>0$ such that $\lambda\cdot\Delta(a_\bullet)$ is not contained in $\Delta(b_\bullet)$, where $\Delta$ is the Newton-Okounkov body of a family. For ideals admitting a Rees package, the same statement holds with $\Delta$ replaced by the closed convex set $\Gamma_R(a_\bullet)$ built from valuation vectors, covering determinantal and Pfaffian ideals among others. If the identification is correct, an infinite family of ideal-containment questions collapses into a single convex-geometric computation.

What carries the argument

The Newton-Okounkov body of a graded family of monomial ideals, $\Delta(a_\bullet)=\bigcup_{k\ge1}(1/k)\operatorname{NP}(a_k)$, is the key object: dilating it by $\lambda$ and asking whether the dilated set leaves $\Delta(b_\bullet)$ mirrors whether $a_{st}$ can escape $b_{rt}$. The polar set $C^\circ=\{a\in\mathbb{R}^n : \langle a,b\rangle\ge1 \text{ for all } b\in C\}$, together with the bipolar lemma for sets absorbing $\mathbb{R}^n_{\ge0}$, converts this dilation threshold into the reciprocal of a minimum pairing $\langle a,b\rangle$. For non-monomial ideals, the Rees package $(B,v,\Gamma)$ substitutes a vector of $B$-monomial valuations for monomial exponents, and $\Gamma_R(a_\bullet)=\bigcup_{k\ge1}(1/k)\operatorname{conv}\{v(b) : b \text{ is a } B\text{-monomial appearing in some } f\in a_k\}$ plays the role of $\Delta(a_\bullet)$. The mechanism throughout is to translate ideal containments into containments of convex sets and then to read off the threshold where one body pokes out of the other.

What would settle it

Take a non-normal monomial ideal $I$ and compare the true asymptotic resurgence $\hat{\rho}(I^{(\bullet)},I_\bullet)$ with the threshold $\sup\{\lambda>0 : \lambda\cdot\operatorname{SP}(I)\not\subseteq\operatorname{NP}(I)\}$; because $\operatorname{NP}(I)=\operatorname{NP}(\bar{I})$, any discrepancy shows the convex formula computes the integral-closure resurgence rather than the actual one, so Theorem 2.7 would fail without a normality-type hypothesis.

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Extended reading notes

Core claim

The central claim is that asymptotic resurgence is a dilation-threshold invariant of convex bodies. Theorem 2.7 states that for graded families $a_\bullet$, $b_\bullet$ of monomial ideals in a polynomial ring, if $R(b_\bullet)$ is Noetherian then $\hat{\rho}(a_\bullet,b_\bullet)=\sup\{\lambda>0 : \lambda\cdot\Delta(a_\bullet)\not\subseteq\Delta(b_\bullet)\}$, and when this set is nonempty the supremum equals $1/\inf\{\langle a,b\rangle : a\in\Delta(a_\bullet), b\in\Delta(b_\bullet)^\circ\}$. The same shape is proved in Theorem 3.6 for arbitrary ideals in a universally Japanese ring once $b$ carries a Rees package $(B,v,\Gamma)$: $\hat{\rho}(a_\bullet,b_\bullet)=\sup\{\lambda>0 : \lambda\cdot\Gamma_R(a_\bullet)\not\subseteq\Gamma\}$, with $b_\bullet$ the ordinary powers of $b$, and Theorem 3.8 extends this to $b$-equivalent families when $a_\bullet$ is a filtration. The paper also shows (Theorem 4.3) that truncating the first family gives a sequence whose supremum and limit are the true resurgence numbers, while truncating the second family need not.

Load-bearing premise

The proof treats a point in the Newton polyhedron or in $\Gamma_R(a_\bullet)$ as a certificate that some element of the ideal really exists there, but these convex sets describe integral closures; the equality therefore rests on the assumption that convex non-containment matches ideal non-containment, which normality or a suitable integral-closure hypothesis would guarantee.

Editorial extensions

If this is right

  • For monomial families with Noetherian second family, asymptotic resurgence is computable from two Newton-Okounkov bodies; when both are rational polyhedra the computation is a linear program.
  • The formula recovers and generalizes the known edge-ideal duality: $\hat{\rho}(a^{(\bullet)},b_\bullet)=1/\min\{\langle a,b\rangle : a\in\operatorname{SP}(a), b\in\operatorname{SP}(b^\vee)\}$ for squarefree monomial ideals.
  • For ideals with Rees packages—determinantal ideals, symmetric determinantal ideals, Pfaffian ideals, and products of Hankel determinantal ideals—the same threshold formula applies when $b_\bullet$ is the family of ordinary powers, so the asymptotic resurgence is again a single convex computation.
  • In the $b$-equivalent filtration case the convex formula forces the equalities $\hat{\rho}(a_\bullet,b_\bullet)=\hat{\rho}(a_\bullet,\overline{b_\bullet})=\rho(a_\bullet,b_\bullet)$, recovering the example value $10/9$ for symbolic-versus-ordinary powers of a specific monomial ideal.
  • Truncating the first family approximates the true resurgence from below, with the supremum and limit agreeing with $\rho$ and $\hat{\rho}$; Example 4.4 shows the analogous statement for the second family is false.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the threshold formula survives without normality assumptions, it would compute the integral-closure resurgence rather than the ordinary one; a direct check on non-normal monomial ideals would separate the two notions.
  • The same dilation-threshold picture likely applies to other asymptotic invariants of pairs, such as skew initial-degree or valuation-growth invariants, by replacing the convex body with a sublevel set of a valuation.
  • Question 2.10 asks whether the Noetherian hypothesis on $R(b_\bullet)$ can be dropped; a natural test is to compare the convex threshold with the limit over truncations of $a_\bullet$, since truncations approximate $a_\bullet$ but not $b_\bullet$.
  • The Rees-package formulation suggests a numerical recipe: approximate $\Gamma_R(a_\bullet)$ by finite unions of convex hulls of valuation vectors, then solve the dilation containment problem to bound the asymptotic resurgence from below.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes to compute the asymptotic resurgence number of a pair of graded families of ideals from convex bodies: Newton-Okounkov bodies for monomial families and Rees-package bodies for invariant ideals. The main results are Theorem 2.7 for monomial families, Theorems 3.6 and 3.8 for ideals with Rees packages, and Theorem 4.3 on truncations. The central claim is that the asymptotic resurgence equals the dilation threshold of one convex body inside another.

Significance. The idea of reducing an infinite family of ideal containments to a convex-geometric computation is attractive and would be valuable if correct. The paper also contains a useful-looking truncation result (Theorem 4.3) and several worked examples. However, the equivalence between ideal non-containment and convex-body non-containment is not valid for non-normal ideals, and the main theorems fail on a simple monomial example. The truncation part of the paper appears independent and may be correct, but the advertised reduction of resurgence to convex bodies is not established.

major comments (3)
  1. [Theorem 2.7(1), proof, p.6] The claimed equality is false. Let R=k[x,y], b=(x^2,y^2), and a_i=m^{2i}. Then a• and b• are graded families of monomial ideals and R(b•) is Noetherian. For every integer S≥1, a_{St}=m^{2St} is not contained in b^t=(x^2,y^2)^t for any t, since x^{2St}∈a_{St} has total degree 2St>2t and therefore cannot lie in b^t. Hence \hatρ(a•,b•)=∞. On the other hand, ∆(a•)=∆(b•)={u∈R^2_{\ge0}:u_1+u_2≥2}, so {λ>0:λ∆(a•)⊄∆(b•)}=(0,1) and its supremum is 1. The proof's step "u_t∈NP(a_{st})\setminus NP(b_{rt})" from x^{u_t}∈a_{st}\setminus b_{rt} is invalid: membership in the Newton polyhedron describes integral closure, not membership in the ideal. For example, u=(t+1,t-1) lies in NP((x^2,y^2)^t) while x^u is not in that ideal.
  2. [Theorem 3.6, proof, p.9] The load-bearing inference "a_{st}\not\subseteq b^{rt} implies a_{st} is not spanned by the B-monomials whose v-values are in rtΓ" is unjustified. Definition 3.2 only says that the integral closure \overline{b^{rt}} is spanned by those B-monomials; the ideal b^{rt} itself can be strictly smaller. The same counterexample applies: for b=(x^2,y^2), Γ=NP(b), and a_i=m^{2i}, each a_t=m^{2t} is spanned by monomials with v-values in tΓ, yet a_t\not\subseteq b^t. Consequently the first inequality in the proof fails and the theorem's RHS is 1 while \hatρ(a•,b•)=∞. This is a central gap, not a local typo.
  3. [Corollaries 2.11, 2.12; Theorems 3.7, 3.8; Corollary 3.10] Because Theorems 2.7 and 3.6 are false as stated, the derived results that rely on them are unsupported. In particular, Theorem 3.8 uses Theorem 3.6 directly to compute \hatρ(a•,b•) for b-equivalent families, and Corollary 3.10 extends this to determinantal, Pfaffian, and Hankel ideals. These applications therefore do not follow from the manuscript's arguments unless additional normality hypotheses are introduced or the integral-closure resurgence is used instead.
minor comments (4)
  1. [Proof of Theorem 2.7, p.6] The phrase "for any integers s,r ∈ 1" should read "s,r ∈ N".
  2. [Definition 3.2, p.8] The displayed sentence "the integral closure I s is spanned" would be clearer with an overline or explicit \overline{I^s}, since the distinction between I^s and its integral closure is the source of the main gap.
  3. [Proof of Theorem 4.2(2), p.12] The argument asserts that the ascending union \bigcup_n ∆(a_{n,•}) is a closed set; an arbitrary ascending union of closed sets need not be closed. This step needs justification or a different argument, though the statement of Theorem 4.2(2) may still be true.
  4. [Example 4.4, p.14] The notation \hatρ(a•,\overline{b•}) is used without prior definition; please define the integral-closure variant explicitly if it is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convex-body formulas are derived from definitions and prior independent theorems, though the proofs contain an integral-closure gap.

full rationale

Walking the claimed derivation chain: Theorem 2.7(1) starts from non-containment a_st ⊄ b_rt, extracts points in NP(a_st) \ NP(b_rt), and uses [22, Theorem 3.4] only to obtain the stable form of the Noetherian family b•; the equality with sup{λ : λ·Δ(a•) ⊄ Δ(b•)} is obtained by rescaling, not by definition. Theorem 2.7(2) is a routine bipolar computation applied to already-constructed bodies. The Rees-package results in Section 3 take Definition 3.2 from [1] as an input: a Rees package is a triple satisfying a spanning condition for integral closures and rational powers, and Theorem 3.6 proves, rather than assumes, that ordinary asymptotic resurgence equals the dilation threshold of Γ_R(a•). The reductions in Theorem 3.7 and Theorem 3.8 use [21, Lemma 4.14] and [21, Theorem 4.8] as lemmas from earlier work of the same group; these are independent prior statements and are not restatements of the target formula. No parameter is fitted to the target invariant, and no uniqueness theorem or hidden ansatz is imported to force the conclusion. The serious issue in the paper is a correctness gap, not circularity: Newton polyhedra and Rees packages encode integral closures, for instance NP(I) = NP(\bar{I}), and Definition 3.2 describes \bar{I^s}, while the proofs of Theorem 2.7(1) and Theorem 3.6 infer ideal containments such as a_{skt} ⊆ b_{rkt} and b_{rt} being the B-span of {b : v(b) ∈ rtΓ} from convex-body containments. For b = (x^2,y^2) and a_i = m^{2i}, the inference that v ∈ NP(a_q) implies x^v ∈ a_q is false, and the analogue in Theorem 3.6 is unsupported without a normality hypothesis. That means the stated theorems may fail for non-normal families, but the conclusions are not equivalent to the inputs by construction. Hence circularity score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted constants. All numerical values in examples (10/9, 2(m-1)/m) are computed from the stated convex bodies, not chosen to match target answers. The axioms that matter are the Noetherianity of the Rees algebra, the unstated integral-closure assumption, and the Rees package spanning property. No new entities are postulated.

assumptions (4)
  • domain assumption The Rees algebra R(b•) is Noetherian, so that [22, Theorem 3.4] applies and the Newton-Okounkov body stabilizes as (1/k)NP(b_k).
    Invoked in Theorem 2.7(1); the authors explicitly leave open whether the formula holds without it (Question 2.10).
  • ad hoc to paper Newton polyhedron membership is treated as ideal membership for monomial ideals, i.e., the ideals are effectively assumed integrally closed.
    In the proof of Theorem 2.7, v in NP(aq) is used to conclude x^v in aq. This is not valid for a general monomial ideal; NP describes the integral closure.
  • ad hoc to paper The Rees package (B,v,Gamma) detects containment for ordinary powers, not only integral closures of powers.
    Theorem 3.6's proof requires a_st not subset b^(rt) to imply a B-monomial outside rt*Gamma; the package guarantees this only for the integral closure of b^(rt).
  • domain assumption R is a Nagata (universally Japanese) algebra, so integral closures behave finitely.
    Assumed in Section 3 to avoid pathologies like non-finite module extensions, as in Nagata's example.

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Pith. "Pith review of Resurgence number and convex body associated to pairs of graded families of ideals." pith.science (2026). https://pith.science/paper/ZRDF6UPR

@misc{pith2026241204417,
  author       = {Pith},
  title        = {Pith review of: Resurgence number and convex body associated to pairs of graded families of ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZRDF6UPR}},
  note         = {Machine review of arXiv:2412.04417}
}
read the original abstract

We discuss how to understand the asymptotic resurgence number of a pair of graded families of ideals from combinatorial data of their associated convex bodies. When the families consist of monomial ideals, the convex bodies being considered are the Newton-Okounkov bodies of the families. When ideals in the second family are classical invariant ideals, for instance, determinantal ideals or ideals of Pfaffians, these convex bodies are constructed from the associated Rees packages.

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Works this paper leans on

34 extracted references · 33 canonical work pages

  1. [34]

    Villarreal, A duality theorem for the ic-resurgence of edge ideals

    R.H. Villarreal, A duality theorem for the ic-resurgence of edge ideals . Eur. J. Comb. 109 (2023), 103656. Tulane University, Department of Mathematics, 6823 St. Cha rles A ve., New Orleans, LA 70118, USA Email address : tha@tulane.edu Department of Mathematics, I.I.T. Madras, Chennai - 600036 , INDIA Email address : jayanav@iitm.ac.in Department of Math...

    work page 2023

  2. [1]

    Bisui, S

    S. Bisui, S. Das, T.H. H` a, and J. Monta˜ no, Rational powers, invariant ideals, and the summation formula. Preprint (2024), arXiv.org:2402.12350 [math.AC]

    arXiv 2024

  3. [2]

    Bisui, E

    S. Bisui, E. Grifo, H.T. H` a, and T.T. Nguy ˜ ˆ en,Chudnovsky’s conjecture and the stable Harbourne-Huneke containment. Trans. Amer. Math. Soc. Ser. B 9 (2022), 371–394

    work page 2022

  4. [3]

    Bisui, E

    S. Bisui, E. Grifo, H.T. H` a, and T.T. Nguy ˜ ˆ en,Demailly’s Conjecture and the containment problem . J. Pure Appl. Algebra 226 (2022), 106863

    work page 2022

  5. [4]

    Bisui and T

    S. Bisui and T. T. Nguy ˜ ˆ en,Chudnovsky’s conjecture and the stable Harbourne-Huneke c ontainment for general points. J. Algebra 649 (2024) 245–269

    work page 2024

  6. [5]

    Bisui and T

    S. Bisui and T. T. Nguy ˜ ˆ en,Lower bounds for Waldschmidt constants and Demailly’s conj ecture for general and very general points. accepted, Collect. Math. (2024)

    work page 2024

  7. [6]

    Bocci and B

    C. Bocci and B. Harbourne, Comparing powers and symbolic powers of ideals . J. Algebraic Geom. 19 (2010), 3, 399–417

    work page 2010

  8. [7]

    Carlini, H.T

    E. Carlini, H.T. H` a, B. Harbourne and A. Van Tuyl, Ideals of powers and powers of ideals: Intersecting Algebra, Geometry and Combinatorics . Lecture Notes of the Unione Matematica Italiana, Vol. 27. Springer International Publishing, 2020

    work page 2020

Show all 34 references
  1. [8]

    Cooper, Robert J

    Susan M. Cooper, Robert J. D. Embree, Huy T` ai H` a, and Andr ew H. Hoefel, Symbolic powers of monomial ideals. Proc. Edinb. Math. Soc. (2), 60(1) (2017), 39–55

    2017

  2. [9]

    Cutkosky, Multiplicities associated to graded families of ideals

    S.D. Cutkosky, Multiplicities associated to graded families of ideals . Algebra Number Theory 7 (2013), no. 9, 2059–2083

    2013

  3. [10]

    Cutkosky, Asymptotic multiplicities of graded families of ideals and linear series

    S.D. Cutkosky, Asymptotic multiplicities of graded families of ideals and linear series. Adv. Math. 264 (2014), 55–113

    2014

  4. [11]

    Dipasquale, C.A

    M. Dipasquale, C.A. Francisco, J. Mermin and J. Schweig, Asymptotic resurgence via integral closures . Trans. Amer. Math. Soc. 372 (2019), no. 9, 6655–6676

    2019

  5. [12]

    Dipasquale and B

    M. Dipasquale and B. Drabkin, On resurgence via asymptotic resurgence . J. Algebra 587 (2021), 64–84

    2021

  6. [13]

    DiPasquale, T.T

    M. DiPasquale, T.T. Nguy ˜ ˆ en, and A. Seceleanu, Duality for asymptotic invariants of graded families , Adv. Math., 430 (2023), 109208

    2023

  7. [14]

    DiPasquale, T.T

    M. DiPasquale, T.T. Nguy ˜ ˆ en, and A. Seceleanu, Duality for asymptotic invariants of graded families: Combinatorial Applications, In progress (2023)

    2023

  8. [15]

    Dumnicki, B

    M. Dumnicki, B. Harbourne, U. Nagel, A. Seceleanu, T. Szember t and H. Tutaj-Gasi´ nska,Resurgences for ideals of special point configurations in PN coming from hyperplane arrangements . J. Algebra 443 (2015), 383–394

    2015

  9. [16]

    Gitler, E

    I. Gitler, E. Reyes and R. H. Villarreal, Blowup algebras of square-free monomial ideals and some lin ks to combinatorial optimization problems. Rocky Mountain J. Math. 39 (2009), no. 1, 71–102

    2009

  10. [17]

    Grifo, C

    E. Grifo, C. Huneke and V. Mukundan, Expected resurgences and symbolic powers of ideals . J. Lond. Math. Soc. 102 (2020), no. 2, 453–469

    2020

  11. [18]

    Grisalde, A

    G. Grisalde, A. Seceleanu and R.H. Villarreal, Rees algebras of filtration of covering polyhedra and integral closure of powers of monomial ideals . Res. Math. Sci. 9:13 (2022), no. 1

    2022

  12. [19]

    Guardo, B

    E. Guardo, B. Harbourne and A. Van Tuyl, Asymptotic resurgences for ideals of positive dimensional subschemes of projective space . Adv. Math. 246 (2013), 114–127

    2013

  13. [20]

    Harbourne, J

    B. Harbourne, J. Kettinger and F. Zimmitti, Extreme values of the resurgence for homogeneous ideals in polynomial rings . J. Pure Appl. Algebra 226 (2022), 16, 106811

    2022

  14. [21]

    H. T. H` a, A. Kumar, H. D. Nguyen and T. T. Nguy ˜ ˆ en,Resurgence number of graded families of ideals. . Preprint, arXiv.org:2308.16410 (2023)

    2023 arXiv

  15. [22]

    H. T. H` a and T. T. Nguy ˜ ˆ en,Newton-Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals . Trans. Amer. Math. Soc. Ser. B 11 (2024), 1065–1097

    2024

  16. [23]

    I. B. Jafarloo, G. Zito, On the containment problem for fat points . J. Commut. Algebra 13(3) (2021), 305–321

    2021

  17. [24]

    Jayanthan, A

    A.V. Jayanthan, A. Kumar and V. Mukundan, On the resurgence and asymptotic resurgence of homo- geneous ideals. Math. Z. 302 (2022), 2407—2434

    2022

  18. [25]

    Jeffries, J

    J. Jeffries, J. Monta˜ no and M. Varbaro, Multiplicities of classical varieties . Proc. London Math. Soc. 110 (2015), no. 4, 1033–1055. 15

    2015

  19. [26]

    Kaveh and A.G

    K. Kaveh and A.G. Khovanskii, Newton-Okounkov bodies, semigroups of integral points, gr aded algebras and intersection theory . Ann. of Math. (2), 176 (2012), no. 2, 925–978

    2012

  20. [27]

    Kaveh and A.G

    K. Kaveh and A.G. Khovanskii, Convex bodies and multiplicities of ideals . Proc. Steklov Inst. Math. 286 (2014), 268–284. Reprint of Tr. Mat. Inst. Steklova 286 (2014), 291–307

    2014

  21. [28]

    Kumar and V

    A. Kumar and V. Mukundan, Symbolic Powers of Classical varieties . Preprint (2024), arXiv.org:2402.18693 [math.AC]

    2024 arXiv

  22. [29]

    Lazarsfeld and M

    R. Lazarsfeld and M. Mustat ¸a, Convex bodies associated to linear series . Ann. Sci. ´Ec. Norm. Sup´ er. (4), 42 (2009), no. 5, 783–835

    2009

  23. [30]

    T. T. Nguy ˜ ˆ en,The initial degree of symbolic powers of Fermat-like ideals of planes and lines arrange- ments. Comm. Algebra, 51(1) (2023), 29–45

    2023

  24. [31]

    T. T. Nguy ˜ ˆ en,Initial degree of symbolic powers of ideals of Fermat configu rations of points. Rocky Mountain J. Math. 53(3) (2023), 859–874

    2023

  25. [32]

    Rockafellar, Convex Analysis

    R.T. Rockafellar, Convex Analysis. Princeton, NJ: Princeton Un iversity Press. pp. 121–125

  26. [33]

    Swanson and C

    I. Swanson and C. Huneke, Integral closure of ideals, rings, a nd modules. London Math. Soc. Lecture Note Ser., 336. Cambridge University Press, Cambridge, 2006, xiv+431 pp

    2006

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